Chapter 17. The Circular Functions Sides and Angles of a Right Triangle

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1 Chapter 17 The Circular Functions SupposeCosmo begins at locationand walks in a counterclockwise direction, always maintaining a tight 0 ft long tether. As Cosmo moves around the circle, how can we describe his location at any given instant? In one sense, we have already answered this question: The measure of S 1 exactly pins down a location on the circle of radius 0 feet. But, we really might prefer a description of the horizontal and vertical coordinates of Cosmo; this would tie in better with the coordinate system we typically use. Solving this problem will require NEW functions, called the circular functions Sides and Angles of a ight Triangle S 3 S 0 feet S 4 S 1 Figure 17.1: Cosmo moves counterclockwise maintaining a tight tether. Where s Cosmo? Example You are preparing to make your final shot at the British ocket Billiard World Championships. The position of your ball is as in Figure 17., and you must play the ball off the left cushion into the lower-right corner pocket, as indicated by the dotted path. For the big money, where should you aim to hit the cushion? Solution. This problem depends on two basic facts. First, the angles of entry and exit between the path the cushion will be equal. Secondly, the two obvious right triangles in this picture are similar triangles. Let x represent the distance from the bottom left corner to the impact point of the ball s path: roperties of similar triangles tell us that the ratios of 4 common sides are equal: = 1. If we solve this equation 5 x x for x, we obtain x = 15 = 3.75 feet x x The billiard table layout. 4 ft 5 ft find this location 4 1 ft this pocket for the big money 6 ft Mathmatically modeling the bank shot. Figure 17.: A pocket billiard banking problem.

2 CHATE 17. THE CICULA FUNCTINS This discussion is enough to win the tourney. But, of course, there are still other questions we can ask about this simple example: What is the angle? That is going to require substantially more work; indeed the bulk of this Chapter! It turns out, there is a lot of mathematical mileage in the idea of studying ratios of sides of right triangles. The first step, which will get the ball rolling, is to introduce new functions whose very definition involves relating sides and angles of right triangles. 17. The Trigonometric atios hypotenuse B A C side adjacent side opposite Figure 17.3: Labeling the sides of a right triangle. From elementary geometry, the sum of the angles of any trianglewill equal180. Given a righttriangle ABC, since one of the angles is 90, the remaining two angles must be acute angles; i.e., angles of measure between 0 and 90. If we specify one of the acute angles in a right triangle ABC, say angle, we can label the three sides using this terminology. We then consider the following three ratios of side lengths, referred to as trigonometric ratios: sin() def = length of side opposite length of hypotenuse (17.1) cos() def = length of side adjacent length of hypotenuse (17.) tan() def = length of side opposite length of side adjacent to. (17.3) For example, we have three right triangles in Figure 17.4; you can verify that the ythagorean Theorem holds in each of the cases. In the left-hand triangle, sin() = 5 1, cos() =, tan() = 5. In the middle triangle, sin() = 1, cos() = 1, tan() = 1. In the right-hand triangle, sin() = 1, cos() = 3, tan() = 1 3. The symbols sin, cos, and tan are abbreviations for the words sine, cosine and tangent, respectively. As we have defined them, the trigonometric ratios depend on the dimensions of the triangle. However, the same ratios are obtained for any right triangle with acute angle. This follows from the properties of similar triangles. Consider Figure Notice ABC and ADE are similar. If we use ABC to compute cos(), then we find cos() = AC. n the other AB hand, if we use ADE, we obtain cos() = AE. Since the ratios of com- AD mon sides of similar triangles must agree, we have cos() = AC AB = AE AD,

3 17.. THE TIGNMETIC ATIS Figure 17.4: Computing trigonometric ratios for selected right triangles. which is what we wanted to be true. The same argument can be used to show that sin() and tan() can be computed using any right triangle with acute angle. Except for some rigged right triangles, it is not easy D to calculate the trigonometric ratios. Before the 1970 s, approximate values of sin(), cos(), tan() were listed B in long tables or calculated using a slide rule. Today, a scientific calculator saves the day on these computations. Most scientific calculators will give an approximation for A the values of the trigonometric ratios. However, it is good C E to keep in mind we can compute the EXACT values of the Figure 17.5: Applying trigonometric ratios when = 0, π, π, π, π radians or, trigonometric ratios to any right triangle. equivalently, when = 0, 30, 45, 60, 90. Angle Trigonometric atio Deg ad sin() cos() tan() π π π π 1 0 Undefined Table 17.1: Exact Trigonometric atios

4 4 CHATE 17. THE CICULA FUNCTINS Some people make a big deal of approximate vs. exact answers; we won t worry about it here, unless we are specifically asked for an exact answer. However, here is something we will make a big deal about:!!! CAUTIN!!! When computing values of cos(), sin(), and tan() on your calculator, make sure you are using the correct angle mode when entering ; i.e. degrees or radians. For example, if = 1, then cos(1 ) = , sin(1 ) = , and tan(1 ) = In contrast, if = 1 radians, then cos(1) = , sin(1) = , and tan(1) = Applications h h cos() h sin() a a tan() Figure 17.6: What do these ratios mean? When confronted with a situation involving a right triangle where the measure of one acute angle and one side are known, we can solve for the remaining sides using the appropriate trigonometric ratios. Here is the key picture to keep in mind: Important Facts (Trigonometric ratios). Given a right triangle, the trigonometric ratios relate the lengths of the sides as shown in Figure Example To measurethe distance across a river for a new bridge, surveyors placed poles at locations A, B and C. The length AB = 100 feet and the measureof the angle ABC is Find the distance to span the river. If the measurement of the angle ABC is only accurate within ±, find the possible error in AC. B C d A Solution. The trigonometric ratio relating these two sides would be the tangent and we can convert into decimal form, arriving at: tan(31 18 ) = tan(31.3 ) = AC BA = d 100 Figure 17.7: The distance therefore d = 60.8feet. spanning a river. This tells us that the bridge needs to span a gap of 60.8 feet. If the measurement of the angle was in error by +, then tan(31 0 ) = tan( ) = and the span is ft. n the other hand, if the measurement of the angle was in error by, then tan(31 16 ) = tan( ) = and the span is 60.7 ft.

5 17.3. ALICATINS 5 Example A plane is flying 000 feet above sea level toward a mountain. The pilot observes the top of the mountain to be 18 above the horizontal, then immediately flies the plane at an angle of 0 above horizontal. The airspeed of the plane is 100 mph. After 5 minutes, the plane is directly above the top of the mountain. How high is the plane above the top of the mountain (when it passes over)? What is the height of the mountain? 000 ft S sealevel Figure 17.8: Flying toward a mountain. T L E Solution. We can compute the hypotenuse of LT by using the speed and time information about the plane: T = (100mph)(5minutes)(1hour/60minutes) = 5 3 miles. The definitions of the trigonometric ratios show: TL = 5 3 sin(0 ) =.850miles, and L = 5 3 cos(0 ) = 7.831miles. With this data, we can now find EL : EL = L tan(18 ) =.544miles. The height of the plane above the peak is TE = TL EL = = 0.306miles = 1,616feet. The elevation of the peak above sea level is given by: eak elevation = plane altitude + EL = S + EL =,000 + (.544)(5,80) = 15,43 feet. Example A Forest Service helicopter needs to determine the width of a deep canyon. While hovering, they measure the angle γ = 48 at position B (see picture), then descend 400 feet to position A and make two measurements of α = 13 (the measure of EAD), β = 53 (the measure of CAD). Determine the width of the canyon to the nearest foot. B 400 ft γ A C β α E canyon Figure 17.9: Finding the width of a canyon. D Solution. We will need to exploit three right triangles in the picture: BCD, ACD, and ACE. ur goal is to compute ED = CD CE, which suggests more than one right triangle will come into play.

6 6 CHATE 17. THE CICULA FUNCTINS The first step is to use BCD and ACD to obtain a system of two equations and two unknowns involving some of the side lengths; we will then solve the system. From the definitions of the trigonometric ratios, CD = (400+ AC )tan(48 ) CD = AC tan(53 ). lugging the second equation into the first and rearranging we get AC = 400tan(48 ) tan(53 ) tan(48 ) =,053feet. lugging this back into the second equation of the system gives CD = (053)tan(53 ) = 74feet. The next step is to relate ACD and ACE, which can now be done in an effective way using the calculations above. Notice that the measure of CAE is β α = 40. We have CE = AC tan(40 ) = (053)tan(40 ) = 1,73feet. As noted above, ED = CD CE =,74 1,73 = 1,001feet is the width of the canyon Circular Functions 0 x S = (x,y) y If Cosmo is located somewhere in the first quadrant of Figure 17.1, represented by the location S, we can use the trigonometric ratios to describe his coordinates. Impose the indicated xy-coordinate system with origin at and extract the pictured right triangle with vertices at and S. The radius is 0 ft. and applying Fact gives S = (x,y) = (0cos(),0sin()). Figure 17.10: Cosmo on a circular path. Unfortunately, we run into a snag if we allow Cosmo to wander into the second, third or fourth quadrant, since then the angle is no longer acute.

7 17.4. CICULA FUNCTINS Are the trigonometric ratios functions? ecall that sin(), cos(), and tan() are defined for acute angles inside a right triangle. We would like to say that these three equations actually define functions where the variableis an angle. Having said this, it is natural to ask if these three equations can be extended to be defined for 1 A unit circle with radius = 1. ANY angle. For example, we need to explain how sin ( ) π 3 is defined. 1 To start, we begin with the unit circle pictured in the xy-coordinate system. Let = be the angle in standard central position shown in Figure If is positive Figure 17.11: Coordinates (resp. negative), we adopt the convention that is of points on the unit circle. swept out by counterclockwise (resp. clockwise) rotation of the initial side. The objective is to find the coordinates of the point in this figure. Notice that each coordinate of (the x-coordinate and the y-coordinate) will depend on the given angle. For this reason, we need to introduce two new functions involving the variable. Definition Let be an angle in standard central position inside the unit circle, as in Figure This angle determines a point on the unit circle. Define two new functions, cos() and sin(), on the domain of all values as follows: cos() sin() def = horizontal x-coordinate of on unit circle def = vertical y-coordinate of on unit circle r = 1 kilometer We refer to sin() and cos() as the basic circular functions. Keep in mind that these functions have variables which are angles (either in degree or radian measure). These functions will be on your calculator. Again, BE CAEFUL to check the angle mode setting on your calculator ( degrees or radians ) before doing a calculation rad sec Michael starts here Figure 17.1: A circular driving track. Example Michael is test driving a vehicle counterclockwise around a desert test track which is circular of radius 1 kilometer. He starts at the location pictured, traveling 0.05 rad. Impose coordinates as pictured. Where is sec Michael located (in xy-coordinates) after 18 seconds? Solution. Let M(t) be the point on the circle of motion representing Michael s location after t seconds and (t) the angle swept out the by Michael after t seconds. Since we are given the angular speed, we get (t) = 0.05t radians. y-axis M(t) = (x(t),y(t)) (t) 0.05 rad sec x-axis Michael starts here Figure 17.13: Modeling Michael s location.

8 8 CHATE 17. THE CICULA FUNCTINS Since the angle (t) is in central standard position, we get M(t) = (cos((t)), sin((t))) = (cos(0.05t), sin(0.05t)). So, after 18 seconds Michael s location will be M(18) = (0.9004, ).!!! CAUTIN!!! Interpreting the coordinates of the point = (cos(), sin()) in Figure only works if the angle is viewed in central standard position. You must do some additional work if the angle is placed in a different position; see the next Example. Angela starts here 0.03 rad sec y-axis r = 1 kilometer 0.05 rad sec x-axis Michael starts here Example Both Angela and Michael are test driving vehicles counterclockwise around a desert test track which is circular of radius 1 kilometer. They start at the locations shown in Figure 17.14(a). Michael is traveling 0.05 rad/sec and Angela is traveling 0.03 rad/sec. Impose coordinates as pictured. Where are the drivers located (in xy-coordinates) after 18 seconds? (a) Angela and Michael on the same test track. Angela starts here 0.03 rad sec α(t) β(t) A(t) y-axis (t) M(t) 0.05 rad sec x-axis Michael starts here (b) Modeling the motion of Angela and Michael. Figure 17.14: Visualizing motion on a circular track. Solution. Let M(t) be the point on the circle of motion representing Michael s location after t seconds. Likewise, let A(t) be the point on the circle of motion representing Angela s location after t seconds. Let (t) be the angle swept out the by Michael and α(t) the angle swept out by Angela after t seconds. Since we are given the angular speeds, we get (t) = 0.05t radians, and α(t) = 0.03t radians. From the previous Example 17.4., M(t) = (cos(0.05t), sin(0.05t)), and M(18) = (0.9004, ). Angela s angle α(t) is NT in central standard position, so we must observe that α(t)+π = β(t), where β(t) is in central standard position: See Figure 17.14(b). We conclude that A(t) = (cos(β(t)),sin(β(t))) = (cos(π+0.03t),sin(π+0.03t)). So, after 18 seconds Angela s location will be A(18) = ( , ).

9 17.4. CICULA FUNCTINS elating circular functions and right triangles If the point on the unit circle is located in the first quadrant, then we can compute cos() and sin() using trigonometric ratios. In general, it s useful to relate right triangles, the unit circle and the circular functions. To describe this connection, given we place it in central standard position in the unit circle, where =. Draw a line through perpendicular to the x-axis, obtaining an inscribed right triangle. Such a right triangle has hypotenuse of length 1, vertical side of length labeled b and horizontal side of length labeled a. There are four cases: See Figure unit circle (radius = 1) sin() cos() Figure 17.15: The point in the first quadrant. b a b a 01 a b a b CASE I CASE II CASE III CASE IV Figure 17.16: ossible positions of on the unit circle. Case I has already been discussed, arriving at cos() = a and sin() = b. In Case II, we can interpret cos() = a,sin() = b. We can reason similarly in the other Cases III and IV, using Figure 17.16, and we arrive at this conclusion: Important Facts (Circular functions and triangles). View as in Figure and form the pictured inscribed right triangles. Then we can interpret cos() and sin() in terms of these right triangles as follows: Case I : cos() = a, sin() = b Case II : cos() = a, sin() = b Case III : cos() = a, sin() = b Case IV : cos() = a, sin() = b

10 C r T 30 CHATE 17. THE CICULA FUNCTINS 17.5 What About ther Circles? S unit circle Figure 17.17: oints on other circles. general circles: = (x,y) What happens if we begin with a circle C r with radius r (possibly different than 1) and want to compute the coordinates of points on this circle? The circularfunctionscan be used to answer this more general question. icture our circle C r centered at the originin the samepicturewithunit circlec 1 andthe angle in standard central position for each circle. As pictured, we can view = = ST. If = (x,y) is our point on the unit circle corresponding to the angle, then the calculation below shows how to compute coordinates on = (cos(),sin()) C 1 x +y = 1 r x +r y = r (rx) +(ry) = r T = (rx,ry) = (rcos(),rsin()) C r. Important Fact Let C r be a circle of radiusrcenteredat the origin and = ST an angle in standard central position for this circle, as in Figure Then the coordinates of T = (rcos(),rsin()). α B U T S y-axis β = π α =.9416 Q A x-axis circle radius = 1 circle radius = circle radius = 3 Figure 17.18: Coordinates of points on circles. Examples Consider the picture below, with = 0.8 radians and α = 0. radians. What are the coordinates of the labeled points? Solution. The angle is in standard central position; α is a central angle, but it is not in standard position. Notice, β = π α =.9416 is an angle in standard central position which locates the same points U,T,S as the angle α. Applying Definition on page 7: = (cos(0.8), sin(0.8)) = (0.6967, ) Q = ( cos(0.8), sin(0.8)) = (1.3934, ) = (3 cos(0.8), 3 sin(0.8)) = (.0901,.151) S = (cos(.9416), sin(.9416)) = ( , ) T = ( cos(.9416), sin(.9416)) = ( 1.960, ) U = (3 cos(.9416), 3 sin(.9416)) = (.9403, ).

11 17.6. THE BASIC CICULA FUNCTIN 31 Example Suppose Cosmo begins at the position in the figure, walking around the circle of radius 0 feet with an angular speed of 4 M counterclockwise. After 3 5 minutes have elapsed, describe Cosmo s precise location. Solution. Cosmo has traveled 3 4 = 1 revolutions. If is 5 5 the angle traveledafter 3 minutes, = ( 1 rev)( ) π radians 5 rev = 4π radians = 15.08radians. By (15.5.1), we have x = 5 0cos ( 4π rad) = feet and y = 0sin ( 4π rad) = feet. Conclude that Cosmo is located at the point S = ( 16.18,11.76). Using (15.1), = 864 = (360 )+144 ; this means that Cosmo walks counterclockwise around the circle two complete revolutions, plus 144. S 0 feet Figure 17.19: Where is Cosmo after 3 minutes? 17.6 ther Basic Circular Function Given any angle, our constructions offer a concrete link between the cosine and sine functions and right triangles inscribed inside the unit circle: See Figure CASE I CASE II CASE III CASE IV Figure 17.0: Computing the slope of a line using the function tan(). The slope of the hypotenuse of these inscribed triangles is just the slope of the line through. Since = (cos(),sin()) and = (0,0): Slope = y x = sin() cos() ; this would be valid as long as cos() 0. This calculation motivates a new circular function called the tangent of by the rule tan() = sin(), provided cos() 0. cos()

12 3 CHATE 17. THE CICULA FUNCTINS The only time cos() = 0 is when the corresponding point on the unit circle has x-coordinate 0. But, this only happens at the positions (0,1) and (0, 1) on the unit circle, corresponding to angles of the form = ± π,±3π,±5π,. These are the cases when the inscribed right triangle would degenerate to having zero width and the line segment becomes vertical. In summary, we then have this general idea to keep in mind: Important Fact The slope of a line = tan(), where is the angle the line makes with the x-axis (or any other horizontal line) Three other commonly used circular functions come up from time to time. The cotangent function y = cot(), the secant function y = sec() and the cosecant function y = csc() are defined by the formulas: sec() = def 1 cos(), def csc() = 1 sin(), def cot() = 1 tan(). Just as with the tangent function, one needs to worry about the values of for which these functions are undefined (due to division by zero). We will not need these functions in this text. Alaska West SeaTac North South Northwest 50 0 East 0 0 Delta (a) The flight paths of three airplanes. W Alaska Q x = 50 N S x = 30 Northwest y = 0 Delta (b) Modeling the paths of each flight. Figure 17.1: Visualizing and modeling departing airplanes. E Example Three airplanes depart SeaTac Airport. A NorthWest flight is heading in a direction 50 counterclockwise from East, an Alaska flight is heading 115 counterclockwise from East and a Delta flight is heading 0 clockwise from East. Find the location of the Northwest flight when it is 0 miles North of SeaTac. Find the location of the Alaska flight when it is 50 miles West of SeaTac. Find the location of the Delta flight when it is 30 miles East of SeaTac. Solution. We impose a coordinate system in Figure 17.1(a), where East (resp. North ) points along the positive x-axis (resp. positive y-axis). To solve the problem, we will find the equation of the three lines representing the flight paths, then determine where they intersect the appropriate horizontal or vertical line. The Northwest and Alaska directions of flight are angles in standard central position; the Deltaflightdirectionwill be 0. Wecan imagine right triangles with their hypotenuses along the directions of flight, then using the tangent function, we have these three immediate conclusions: slope NW line = tan(50 ) = 1.19, slope Alaska line = tan(115 ) =.14, and slope Delta line = tan( 0 ) =

13 17.6. THE BASIC CICULA FUNCTIN 33 All three flight paths pass through the origin (0,0) of our coordinate system, so the equations of the lines through the flight paths will be: NW flight : y = 1.19x, Alaska flight : y =.14x, Delta flight : y = 0.364x. The Northwest flight is 0 miles North of SeaTac when y = 0; plugging into the equation of the line of flight gives 0 = 1.19x, so x = and the plane location will be = (16.81,0). Similarly, the Alaska flight is 50 miles West of SeaTac when x = 50; plugging into the equation of the line of flight gives y = (.14)( 50) = 107 and the plane location will be Q = ( 50,107). Finally, check that the Delta flight is at = (30, 10.9) when it is 30 miles East of SeaTac.

14 34 CHATE 17. THE CICULA FUNCTINS 17.7 Exercises roblem John has been hired to design an exciting carnival ride. Tiff, the carnival owner, has decided to create the world s greatest ferris wheel. Tiff isn t into math; she simply has a vision and has told John these constraints on her dream: (i) the wheel should rotate counterclockwise with an angular speed of 1 M; (ii) the linear speed of a rider should be 00 mph; (iii) the lowest point on the ride should be 4feet above the level ground. ecall, we worked on this in Exercise forest) as a potential landing site, but are uncertain whether it is wide enough. They make two measurements from A (see picture) finding α = 5 o and β = 54 o. They rise vertically 100 feet to B and measure γ = 47 o. Determine the width of the clearing to the nearest foot. 100 feet B γ A β α 1 M C E clearing D 4 feet (a) Impose a coordinate system and find the coordinates T(t) = (x(t),y(t)) of Tiff at time t seconds after she starts the ride. (b) Tiff becomes a human missile after 6 seconds on the ride. Find Tiff s coordinates the instant she becomes a human missile. (c) Find the equation of the tangential line along which Tiff travels the instant she becomes a human missile. Sketch a picture indicating this line and her initial direction of motion along it when the seat detaches. roblem 17.. (a) Find the equation of a line passing through the point (-1,) and making an angle of 13 o with the x-axis. (Note: There are two answers; find them both.) (b) Find the equation of a line making an angle of 8 o with the y-axis and passing through the point (1,1). (Note: There are two answers; find them both.) roblem Marla is running clockwise around a circular track. She runs at a constant speed of 3 meters per second. She takes 46 seconds to complete one lap of the track. From her starting point, it takes her 1 seconds to reach the northermost point of the track. Impose a coordinate system with the center of the track at the origin, and the northernmost point on the positive y-axis. (a) Give Marla s coordinates at her starting point. (b) Give Marla s coordinates when she has been running for 10 seconds. (c) Give Marla s coordinates when she has been running for seconds. roblem A merry-go-round is rotating at the constant angular speed of 3 M counterclockwise. The platform of this ride is a circular disc of radius 4 feet. You jump onto the ride at the location pictured below. rotating 3 M jump on here roblem The crew of a helicopter needs to land temporarily in a forest and spot a flat horizontal piece of ground (a clearing in the

15 17.7. EXECISES 35 (a) If = 34 o, then what are your xycoordinates after 4 minutes? (b) If = 0 o, then what are your xycoordinates after 45 minutes? (c) If = 14 o, then what are your xycoordinates after 6 seconds? Draw an accurate picture of the situation. (d) If =.1 rad, then what are your xy-coordinates after hours and 7 seconds? Draw an accurate picture of the situation. (e) If =.1 rad, then what are your xycoordinates after 5 seconds? Draw an accurate picture of the situation. roblem Shirley is on a ferris wheel which spins at the rate of 3. revolutions per minute. The wheel has a radius of 45 feet, and the center of the wheel is 59 feet above the ground. After the wheel starts moving, Shirley takes 16 seconds to reach the top of the wheel. How high above the ground is she when the wheel has been moving for 9 minutes? roblem The top of the Boulder Dam has an angle of elevation of 1. radians from a point on the Colorado iver. Measuring the angle of elevation to the top of the dam from a point 155 feet farther down river is 0.9 radians; assume the two angle measurements are taken at the same elevation above sea level. How high is the dam? downriver dam relatively flat area nearby the tower (not necessarily the same altitude as the bottom of the tower), and standing some unknown distance away from the tower, you make three measurements all at the same height above sea level. You observe that the top of the old tower makes an angle of 39 above level. You move 110 feet away from the original measurement and observe that the old top of the tower now makes an angle of 34 above level. Finally, after the new construction is complete, you observe that the new top of the tower, from the same point as the second measurement was made, makes an angle of 40 above the horizontal. All three measurements are made at the same height above sea level and are in line with the tower. Find the height of the addition to the tower, to the nearest foot. roblem Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost point. They both run counterclockwise. Alexandra runs at 4 meters per second, and will take exactly minutes to catch up to Charlie. Impose a coordinate system, and give the x- and y-coordinates of Charlie after one minute of running. roblem George and aula are running around a circular track. George starts at the westernmost point of the track, and aula starts at the easternmost point. The illustration below shows their starting positions and running directions. They start running toward each other at constant speeds. George runs at 9 feet per second. aula takes 50 seconds to run a lap of the track. George and aula pass each other after 11 seconds. 155 ft a N roblem A radio station obtains a permit to increase the height of their radio tower on Queen Anne Hill by no more than 100 feet. You are the head of the Queen Anne Community Group and one of your members asks you to make sure that the radio station does not exceed the limits of the permit. After finding a George aula After running for 3 minutes, how far east of his starting point is George?

16 36 CHATE 17. THE CICULA FUNCTINS roblem A kite is attached to 300 feet of string, which makes a 4 degree angle with the level ground. The kite pilot is holding the string 4 feet above the ground. kite (d) What are the coordinates of the bug after 1 second? After 0 seconds? After 3 seconds? After seconds? ω=4π/9rad/sec bug lands here 1. rad 4 o in 4 feet ground level (a) How high above the ground is the kite? (b) Suppose that power lines are located 50 feet in front of the kite flyer. Is any portion of the kite or string over the power lines? roblem In the pictures below, a bug has landed on the rim of a jelly jar and is moving around the rim. The location where the bug initially lands is described and its angular speed is given. Impose a coordinate system with the origin at the center of the circle of motion. In each of the cases, answer these questions: (a) Find an angle 0 in standard central position that gives the bugs initial location. (In some cases, this is the angle given in the picture; in other cases, you will need to do something.) (b) The location angle of the bug at time t is given by the formula (t) = 0 +ωt. lug in the values for 0 and ω to explicitly obtain a formula for (t). (c) Find the coordinates of the bug at time t. bug lands here ω=4π/9rad/sec ω= 4π/9rad/sec bug lands here 0.5 rad in in